## (2000)

### Abstract

Chern numbers for singular varieties and elliptic homology By Burt Totaro A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so. Here we use ideas from minimal model theory to define some characteristic numbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was suggested by Goresky and MacPherson as a next natural problem after the definition of intersection homology [11]. We find that only a subspace of the Chern numbers can be defined for singular varieties. A convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism, whereas a singular variety can at most have a fundamental class in a weaker homology theory, elliptic homology. We use this idea to give an algebro-geometric definition of elliptic homology: “complex bordism modulo flops equals elliptic homology.” This paper was inspired by some questions asked by Jack Morava. The descriptions of elliptic homology given by Gerald Höhn [13] were also an important

### Citations

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215 |
The projectivity of the moduli space of stable curves
- Knudsen, Mumford
(Show Context)
Citation Context ...re Mg,r is the Knudsen-Deligne-Mumford moduli stack of r-pointed stable curves of genus g, which comes with r line bundles ψ1,...,ψr representing the cotangent line of the curve at the r given points =-=[15]-=-. The line bundle ψ1 on M1,2 is not the pullback of the line bundle ψ1 on M1,1 by the projection π : M1,2 → M1,1, forgetting the second point; instead, we have ψ1 = π ∗ ψ1 + D0,1, where D0,1 is the di... |

181 |
The Theory of Jacobi Forms
- Eichler, Zagier
- 1985
(Show Context)
Citation Context ...)[[q]]. Moreover, for X an SU-manifold of complex dimension n, ϕ(X) is in fact a Jacobi form of weight n, by Höhn [13]. Jacobi forms are generalizations of modular forms defined by Eichler and Zagier =-=[8]-=-, although we use a slight variant of their definition as we will explain later in this section. Just as modular forms (of level 1) are exactly sections of powers of a certain line bundle ψ1 on the co... |

117 | Intersection homology - Goresky, Macpherson - 1983 |

30 |
Riemann–Roch and topological K-theory for singular varieties
- Baum, Fulton, et al.
- 1979
(Show Context)
Citation Context ...variety X. It is equal to the class of Ω p X for X smooth, and it satisfies f∗χp(X) = χp(Y ) for an IH-small resolution f : X → Y . Finally, we can apply Baum-Fulton-MacPherson’s natural homomorphism =-=[4]-=- from the Grothendieck group G0X of coherent sheaves to topological Khomology K top 0 X, followed by the homological version of the Chern character hh : K top 0 X → H∗(X,Q) (also described in [4]), to... |

29 |
Generalized elliptic genera and Baker-Akhiezer functions
- Krichever
- 1990
(Show Context)
Citation Context ... invariant of compact complex manifolds, is unchanged under “classical flops” if and only if it is a linear combination of the coefficients of the complex elliptic genus studied by Krichever and Höhn =-=[19]-=-, [13]. This elliptic genus can be viewed as a power series associated to any compact complex manifold, the coefficients of the series being certain fixed linear combinations of the Chern numbers of t... |

25 |
Introduction to the minimal model program, Algebraic Geometry
- Kawamata, Matsuda, et al.
- 1987
(Show Context)
Citation Context ...ne extension of the Chern number ck 1χn−k i to singular n-folds, and we can do this for those varieties which have a relative canonical model; again, the minimal model conjecture (Conjecture 0-4-4 in =-=[14]-=-) would imply that every variety has a relative canonical model (Theorem 3-3-1 in [14]). We recall some of the relevant definitions. A relative canonical model for a variety Y is defined, starting fro... |

24 |
On bordism theory of manifolds with singularities
- Baas
- 1973
(Show Context)
Citation Context ...he results above, the Sullivan-Baas method of bordism with singularities produces a multiplicative cohomology theory which is a module over MSU ⊗Z[1/2] and which has coefficient ring Z[1/2][x2,x3,x4] =-=[2]-=-, [24]. (The Sullivan-Baas method gives a cohomology theory with the coefficient ring we want because the ideal I is defined by a regular sequence in the ring MSU∗ ⊗ Z[1/2] = Z[1/2][x2,x3,x4,...], as ... |

24 |
Topological methods in algebraic geometry” (3rd edition
- Hirzebruch
- 1966
(Show Context)
Citation Context ...mplex projective n-folds [23].) For example, the Euler characteristic, the signature, and the Todd genus, for smooth projective varieties of a given dimension, are Chern numbers, thanks to Hirzebruch =-=[12]-=-. Let us define an IH-small resolution of a singular variety Y to be a resolution of singularities f : X → Y such that for every i ≥ 1, the set of points y ∈ Y such that dim(f −1 (y)) = i has codimens... |

19 |
Courbes elliptiques: Formulaire (d’après
- Deligne
(Show Context)
Citation Context ...ng of (level 1) modular forms over any Z[1/6]-algebra R is the polynomial ring R[g2,g3], where gi is the Eisenstein series of weight 2i, i ≥ 2. The ring is more complicated in characteristics 2 and 3 =-=[6]-=-. By the methods of Deligne’s paper, one can also compute the ring of Jacobi forms, in the above sense. For any Z[1/6]-algebra R, one gets the polynomial ring R[x,y,g2], where x is the Weierstrass p-f... |

17 |
On analytic surfaces with double points
- Atiyah
- 1958
(Show Context)
Citation Context ...o the same weakly complex manifold M. Then in the bordism group MU∗. A ∪M −B + B ∪M −C + C ∪M −A = 0 Proof. Let H denote a hexagon in R 2 (thus H is homeomorphic to the disk). Let W be the union of A×=-=[0,1]-=-, B ×[0,1], C ×[0,1], and M ×H modulo A M 1 0 A M �❅ ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅❅ � � M H M 0 1 B C 1 0 M M B C the identifications pictured here. Then W is a compact weakly complex manifold whose bound... |

16 |
Relèvement de cycles algébriques et homomorphismes associés en homologie d’intersection
- Barthel, Brasselet, et al.
- 1995
(Show Context)
Citation Context ... as Goresky and MacPherson conjectured, all algebraic cycles, and in particular the homology Chern classes, lift rationally to intersection homology, by Barthel, Brasselet, Fieseler, Gabber, and Kaup =-=[3]-=-; but those lifts are not unique (see the comments on this problem in [3, p. 158]), and we will not use that approach. Clearly, if a given Chern number can be defined for singular varieties in the abo... |

16 |
Complex de De Rham filtré d’une variété singulière
- Bois
- 1981
(Show Context)
Citation Context ...a generalization of the sheaf of p-forms on a smooth variety. (A different generalization of the sheaf of p-forms on a smooth variety to an object in the derived category was found earlier by du Bois =-=[7]-=-, related to ordinary cohomology rather than intersection cohomology. Saito was partly inspired by du Bois’s work.) The filtration F is preserved under proper pushforward in a precise sense [33, p. 27... |

14 |
The Structure of Algebraic Threefolds: an Introduction to Mori’s Program
- Kollar
- 1987
(Show Context)
Citation Context ...a point in Y . Again, some references are [31, Th. 0.2] for the toric case, [17, p. 121] for the definition of terminal singularities, and [14] for the most detailed development of the theory. Kollár =-=[16]-=- gives a good introductory survey, and his more recent survey [17] is also very useful. If X is a projective IH-small resolution of a variety Y , then X is smooth and hence has Q-factorial terminal si... |

12 |
Komplexe elliptische Geschlechter und S1-äquivariante Kobordismustheorie
- Höhn
- 1991
(Show Context)
Citation Context ... elliptic homology: “complex bordism modulo flops equals elliptic homology.” This paper was inspired by some questions asked by Jack Morava. The descriptions of elliptic homology given by Gerald Höhn =-=[13]-=- were also an important influence. Thanks to Dave Bayer, Mike Stillman, John Stembridge, Sheldon Katz, and Stein Stromme for their computer algebra programs Macaulay, SF, and Schubert, which helped in... |

8 |
HP 2 -bundles and elliptic homology
- Kreck, Stolz
- 1993
(Show Context)
Citation Context ...x4]. These results are analogous to the results of Kreck and Stolz, describing the kernel of the Ochanine genus on MSpin ∗ in terms of HP 2 -bundles, except that for now we work away from the prime 2 =-=[18]-=-. Remarks. (1) For integral questions such as this, it seems more natural to work with the ring MSU∗ rather than MU∗, for example because the image of MU∗ under the complex elliptic genus is not finit... |

6 |
flops, minimal models, etc
- Flips
- 1991
(Show Context)
Citation Context ... a relative canonical model. The extension is compatible with IH-small resolutions when they exist. The minimal model conjecture would imply that every singular variety has a relative canonical model =-=[17]-=-. For example, for n ≤ 4 the twisted χy genus includes all Chern numbers for n-folds. So we know that all Chern numbers for n-folds with n ≤ 4 take the same value on any two IH-small resolutions of a ... |

2 |
Sur l’homologie d’intersection et les classes de Chern des variétés singulières (espaces de Thom, exemples de J.-L. Verdier et M. Goresky
- Brasselet, Gonzalez-Sprinberg
- 1984
(Show Context)
Citation Context ...pposed to ordinary homology, since even the simplest Chern polynomials, namely the Chern classes, can be different in two different IH-small resolutions of the same singular variety, as Verdier found =-=[5]-=-. Also, there is a natural integral version of the question we have been considering rationally. One could try to compute the quotient ring of the complex bordism ring MU∗ by flops, but this ring seem... |

1 |
and bibliography on intersection homology
- Problems
- 1984
(Show Context)
Citation Context ...umbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was suggested by Goresky and MacPherson as a next natural problem after the definition of intersection homology =-=[11]-=-. We find that only a subspace of the Chern numbers can be defined for singular varieties. A convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex... |